Multiresolution analysis

A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ironing method) and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James L. Crowley. A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ironing method) and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James L. Crowley. A multiresolution analysis of the Lebesgue space L 2 ( R ) {displaystyle L^{2}(mathbb {R} )} consists of a sequence of nested subspaces that satisfies certain self-similarity relations in time-space and scale-frequency, as well as completeness and regularity relations. In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to Ingrid Daubechies. Assuming the scaling function has compact support, then V 0 ⊂ V − 1 {displaystyle V_{0}subset V_{-1}} implies that there is a finite sequence of coefficients a k = 2 ⟨ ϕ ( x ) , ϕ ( 2 x − k ) ⟩ {displaystyle a_{k}=2langle phi (x),phi (2x-k) angle } for | k | ≤ N {displaystyle |k|leq N} , and a k = 0 {displaystyle a_{k}=0} for | k | > N {displaystyle |k|>N} , such that Defining another function, known as mother wavelet or just the wavelet one can show that the space W 0 ⊂ V − 1 {displaystyle W_{0}subset V_{-1}} , which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to V 0 {displaystyle V_{0}} inside V − 1 {displaystyle V_{-1}} . Or put differently, V − 1 {displaystyle V_{-1}} is the orthogonal sum (denoted by ⊕ {displaystyle oplus } ) of W 0 {displaystyle W_{0}} and V 0 {displaystyle V_{0}} . By self-similarity, there are scaled versions W k {displaystyle W_{k}} of W 0 {displaystyle W_{0}} and by completeness one has